A contradiction is the simultaneous acceptance and denial of a proposition or statement. "It was the best of times and the worst of times" is a poetic example: the contradiction is being used to enhance the poetic nature of the piece.
According to J.Sarfati, a contradiction is defined as the conjunction of the affirmation and denial of a premise, in the same time, place, and sense (i.e. p and not-p, or in symbolic form, p.~p).
It is often held that from a contradiction anything can be inferred. This can be demonstrated:
- Take any contradiction, “A” and “Not A”, and take it to be true.
- Take anything that you want to prove, “P”
- Now because “A and Not A” is true, “A” is true, by virtue of the simplification rule.
- “A or P” is true just so long as not both A and P are false.
- So, As “A” is true, “A or P” must be true.
- If “A or P” is true, then one of “A” or “P” must be true.
- As “Not A” is true then “A” is false which means that the remaining term in “A or P” must be true.
- “P”, absolutely anything you like, is true.